Definition:Cantor Set

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Definition

As a Limit of Intersections

Define, for $n \in \N$, subsequently:

$\map k n := \dfrac {3^n - 1} 2$
$\displaystyle A_n := \bigcup_{i \mathop = 1}^{\map k n} \openint {\frac {2 i - 1} {3^n} } {\frac {2 i} {3^n} }$

Since $3^n$ is always odd, $\map k n$ is always an integer, and hence the union will always be perfectly defined.


Consider the closed interval $\closedint 0 1 \subset \R$.

Define:

$\mathcal C_n := \closedint 0 1 \setminus A_n$

The Cantor set $\mathcal C$ is defined as:

$\displaystyle \mathcal C = \bigcap_{n \mathop = 1}^\infty \mathcal C_n$


From Ternary Representation

Consider the closed interval $\closedint 0 1 \subset \R$.

The Cantor set $\mathcal C$ consists of all the points in $\closedint 0 1$ which can be expressed in base $3$ without using the digit $1$.


From Representation of Ternary Expansions, if any number has two different ternary representations, for example:

$\dfrac 1 3 = 0.10000 \ldots = 0.02222$

then at most one of these can be written without any $1$'s in it.

Therefore this representation of points of $\mathcal C$ is unique.


As a Limit of a Decreasing Sequence

Let $\map {I_c} \R$ denote the set of all closed real intervals.


Define the mapping $t_1: \map {I_c} \R \to \map {I_c} \R$ by:

$\map {t_1} {\closedint a b} := \closedint a {\dfrac 1 3 \paren {a + b} }$

and similarly $t_3: \map {I_c} \R \to \map {I_c} \R$ by:

$\map {t_3} {\closedint a b} := \closedint {\dfrac 2 3 \paren {a + b} } b$

Note in particular how:

$\map {t_1} {\closedint a b} \subseteq \closedint a b$
$\map {t_3} {\closedint a b} \subseteq \closedint a b$


Subsequently, define inductively:

$S_0 := \set {\closedint 0 1}$
$S_{n + 1} := \map {t_1} {C_n} \cup \map {t_3} {C_n}$

and put, for all $n \in \N$:

$C_n := \displaystyle \bigcup S_n$

Note that $C_{n + 1} \subseteq C_n$ for all $n \in \N$, so that this forms a decreasing sequence of sets.


Then the Cantor set $\mathcal C$ is defined as its limit, that is:

$\mathcal C := \displaystyle \bigcap_{n \mathop \in \N} C_n$


These definitions are all equivalent, as shown on Equivalence of Definitions of Cantor Set.


Also known as

Some sources refer to the Cantor set as Cantor's discontinuum.

Some sources refer to this specifically as Cantor's middle third set or Cantor's ternary set.


Also defined as

Some sources define a Cantor set as any set topologically equivalent to this set.


Also see

  • Results about the Cantor set can be found here.


Source of Name

This entry was named for Georg Cantor.


Sources